\(\int \frac {1}{(-2+2 x^2-3 x^4)^{3/2}} \, dx\) [283]

Optimal result

Integrand size = 16, antiderivative size = 262 \[ \int \frac {1}{\left (-2+2 x^2-3 x^4\right )^{3/2}} \, dx=-\frac {x \left (4+3 x^2\right )}{20 \sqrt {-2+2 x^2-3 x^4}}-\frac {3 x \sqrt {-2+2 x^2-3 x^4}}{20 \left (\sqrt {6}+3 x^2\right )}-\frac {\sqrt [4]{3} \left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2-2 x^2+3 x^4}{\left (2+\sqrt {6} x^2\right )^2}} E\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{12} \left (6+\sqrt {6}\right )\right )}{10\ 2^{3/4} \sqrt {-2+2 x^2-3 x^4}}+\frac {\sqrt [4]{3} \left (1-\sqrt {6}\right ) \left (2+\sqrt {6} x^2\right ) \sqrt {\frac {2-2 x^2+3 x^4}{\left (2+\sqrt {6} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right ),\frac {1}{12} \left (6+\sqrt {6}\right )\right )}{20\ 2^{3/4} \sqrt {-2+2 x^2-3 x^4}} \] Output:

-1/20*x*(3*x^2+4)/(-3*x^4+2*x^2-2)^(1/2)-3*x*(-3*x^4+2*x^2-2)^(1/2)/(20*6^ 
(1/2)+60*x^2)-1/20*3^(1/4)*(2+6^(1/2)*x^2)*((3*x^4-2*x^2+2)/(2+6^(1/2)*x^2 
)^2)^(1/2)*EllipticE(sin(2*arctan(1/2*3^(1/4)*2^(3/4)*x)),1/6*(18+3*6^(1/2 
))^(1/2))*2^(1/4)/(-3*x^4+2*x^2-2)^(1/2)+1/40*3^(1/4)*(1-6^(1/2))*(2+6^(1/ 
2)*x^2)*((3*x^4-2*x^2+2)/(2+6^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan 
(1/2*3^(1/4)*2^(3/4)*x),1/6*(18+3*6^(1/2))^(1/2))*2^(1/4)/(-3*x^4+2*x^2-2) 
^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.45 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\left (-2+2 x^2-3 x^4\right )^{3/2}} \, dx=\frac {-3 \sqrt {-\frac {i}{i+\sqrt {5}}} x \left (4+3 x^2\right )-\sqrt {3} \left (-i+\sqrt {5}\right ) \sqrt {\frac {i+\sqrt {5}-3 i x^2}{i+\sqrt {5}}} \sqrt {\frac {-i+\sqrt {5}+3 i x^2}{-i+\sqrt {5}}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {3 i}{i+\sqrt {5}}} x\right )|\frac {i+\sqrt {5}}{i-\sqrt {5}}\right )+\sqrt {3} \left (5 i+\sqrt {5}\right ) \sqrt {\frac {i+\sqrt {5}-3 i x^2}{i+\sqrt {5}}} \sqrt {\frac {-i+\sqrt {5}+3 i x^2}{-i+\sqrt {5}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {3 i}{i+\sqrt {5}}} x\right ),\frac {i+\sqrt {5}}{i-\sqrt {5}}\right )}{60 \sqrt {-\frac {i}{i+\sqrt {5}}} \sqrt {-2+2 x^2-3 x^4}} \] Input:

Integrate[(-2 + 2*x^2 - 3*x^4)^(-3/2),x]
 

Output:

(-3*Sqrt[(-I)/(I + Sqrt[5])]*x*(4 + 3*x^2) - Sqrt[3]*(-I + Sqrt[5])*Sqrt[( 
I + Sqrt[5] - (3*I)*x^2)/(I + Sqrt[5])]*Sqrt[(-I + Sqrt[5] + (3*I)*x^2)/(- 
I + Sqrt[5])]*EllipticE[I*ArcSinh[Sqrt[(-3*I)/(I + Sqrt[5])]*x], (I + Sqrt 
[5])/(I - Sqrt[5])] + Sqrt[3]*(5*I + Sqrt[5])*Sqrt[(I + Sqrt[5] - (3*I)*x^ 
2)/(I + Sqrt[5])]*Sqrt[(-I + Sqrt[5] + (3*I)*x^2)/(-I + Sqrt[5])]*Elliptic 
F[I*ArcSinh[Sqrt[(-3*I)/(I + Sqrt[5])]*x], (I + Sqrt[5])/(I - Sqrt[5])])/( 
60*Sqrt[(-I)/(I + Sqrt[5])]*Sqrt[-2 + 2*x^2 - 3*x^4])
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1405, 27, 1511, 27, 1416, 1509}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{\left (-3 x^4+2 x^2-2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1405

\(\displaystyle -\frac {1}{40} \int \frac {6 \left (2-x^2\right )}{\sqrt {-3 x^4+2 x^2-2}}dx-\frac {x \left (3 x^2+4\right )}{20 \sqrt {-3 x^4+2 x^2-2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{20} \int \frac {2-x^2}{\sqrt {-3 x^4+2 x^2-2}}dx-\frac {x \left (3 x^2+4\right )}{20 \sqrt {-3 x^4+2 x^2-2}}\)

\(\Big \downarrow \) 1511

\(\displaystyle -\frac {3}{20} \left (\frac {1}{3} \left (6-\sqrt {6}\right ) \int \frac {1}{\sqrt {-3 x^4+2 x^2-2}}dx+\sqrt {\frac {2}{3}} \int \frac {2-\sqrt {6} x^2}{2 \sqrt {-3 x^4+2 x^2-2}}dx\right )-\frac {x \left (3 x^2+4\right )}{20 \sqrt {-3 x^4+2 x^2-2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{20} \left (\frac {1}{3} \left (6-\sqrt {6}\right ) \int \frac {1}{\sqrt {-3 x^4+2 x^2-2}}dx+\frac {\int \frac {2-\sqrt {6} x^2}{\sqrt {-3 x^4+2 x^2-2}}dx}{\sqrt {6}}\right )-\frac {x \left (3 x^2+4\right )}{20 \sqrt {-3 x^4+2 x^2-2}}\)

\(\Big \downarrow \) 1416

\(\displaystyle -\frac {3}{20} \left (\frac {\int \frac {2-\sqrt {6} x^2}{\sqrt {-3 x^4+2 x^2-2}}dx}{\sqrt {6}}+\frac {\left (6-\sqrt {6}\right ) \left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4-2 x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right ),\frac {1}{12} \left (6+\sqrt {6}\right )\right )}{6 \sqrt [4]{6} \sqrt {-3 x^4+2 x^2-2}}\right )-\frac {x \left (3 x^2+4\right )}{20 \sqrt {-3 x^4+2 x^2-2}}\)

\(\Big \downarrow \) 1509

\(\displaystyle -\frac {3}{20} \left (\frac {\left (6-\sqrt {6}\right ) \left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4-2 x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right ),\frac {1}{12} \left (6+\sqrt {6}\right )\right )}{6 \sqrt [4]{6} \sqrt {-3 x^4+2 x^2-2}}+\frac {\frac {2^{3/4} \left (\sqrt {6} x^2+2\right ) \sqrt {\frac {3 x^4-2 x^2+2}{\left (\sqrt {6} x^2+2\right )^2}} E\left (2 \arctan \left (\sqrt [4]{\frac {3}{2}} x\right )|\frac {1}{12} \left (6+\sqrt {6}\right )\right )}{\sqrt [4]{3} \sqrt {-3 x^4+2 x^2-2}}+\frac {2 \sqrt {-3 x^4+2 x^2-2} x}{\sqrt {6} x^2+2}}{\sqrt {6}}\right )-\frac {x \left (3 x^2+4\right )}{20 \sqrt {-3 x^4+2 x^2-2}}\)

Input:

Int[(-2 + 2*x^2 - 3*x^4)^(-3/2),x]
 

Output:

-1/20*(x*(4 + 3*x^2))/Sqrt[-2 + 2*x^2 - 3*x^4] - (3*(((2*x*Sqrt[-2 + 2*x^2 
 - 3*x^4])/(2 + Sqrt[6]*x^2) + (2^(3/4)*(2 + Sqrt[6]*x^2)*Sqrt[(2 - 2*x^2 
+ 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticE[2*ArcTan[(3/2)^(1/4)*x], (6 + Sqrt 
[6])/12])/(3^(1/4)*Sqrt[-2 + 2*x^2 - 3*x^4]))/Sqrt[6] + ((6 - Sqrt[6])*(2 
+ Sqrt[6]*x^2)*Sqrt[(2 - 2*x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*A 
rcTan[(3/2)^(1/4)*x], (6 + Sqrt[6])/12])/(6*6^(1/4)*Sqrt[-2 + 2*x^2 - 3*x^ 
4])))/20
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1405

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 
- 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) 
), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(b^2 - 2*a*c + 2*(p + 1)*( 
b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr 
eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.74 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {x \left (3 x^{2}+4\right )}{20 \sqrt {-3 x^{4}+2 x^{2}-2}}-\frac {3 \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2-2 i \sqrt {5}}\, x}{2}, \frac {\sqrt {-6+3 i \sqrt {5}}}{3}\right )}{5 \sqrt {2-2 i \sqrt {5}}\, \sqrt {-3 x^{4}+2 x^{2}-2}}+\frac {6 \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {2-2 i \sqrt {5}}\, x}{2}, \frac {\sqrt {-6+3 i \sqrt {5}}}{3}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {2-2 i \sqrt {5}}\, x}{2}, \frac {\sqrt {-6+3 i \sqrt {5}}}{3}\right )\right )}{5 \sqrt {2-2 i \sqrt {5}}\, \sqrt {-3 x^{4}+2 x^{2}-2}\, \left (2+2 i \sqrt {5}\right )}\)	\(237\)
default	\(\frac {-\frac {1}{5} x -\frac {3}{20} x^{3}}{\sqrt {-3 x^{4}+2 x^{2}-2}}-\frac {3 \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2-2 i \sqrt {5}}\, x}{2}, \frac {\sqrt {-6+3 i \sqrt {5}}}{3}\right )}{5 \sqrt {2-2 i \sqrt {5}}\, \sqrt {-3 x^{4}+2 x^{2}-2}}+\frac {6 \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {2-2 i \sqrt {5}}\, x}{2}, \frac {\sqrt {-6+3 i \sqrt {5}}}{3}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {2-2 i \sqrt {5}}\, x}{2}, \frac {\sqrt {-6+3 i \sqrt {5}}}{3}\right )\right )}{5 \sqrt {2-2 i \sqrt {5}}\, \sqrt {-3 x^{4}+2 x^{2}-2}\, \left (2+2 i \sqrt {5}\right )}\)	\(238\)
elliptic	\(\frac {-\frac {1}{5} x -\frac {3}{20} x^{3}}{\sqrt {-3 x^{4}+2 x^{2}-2}}-\frac {3 \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2-2 i \sqrt {5}}\, x}{2}, \frac {\sqrt {-6+3 i \sqrt {5}}}{3}\right )}{5 \sqrt {2-2 i \sqrt {5}}\, \sqrt {-3 x^{4}+2 x^{2}-2}}+\frac {6 \sqrt {1-\left (\frac {1}{2}-\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}+\frac {i \sqrt {5}}{2}\right ) x^{2}}\, \left (\operatorname {EllipticF}\left (\frac {\sqrt {2-2 i \sqrt {5}}\, x}{2}, \frac {\sqrt {-6+3 i \sqrt {5}}}{3}\right )-\operatorname {EllipticE}\left (\frac {\sqrt {2-2 i \sqrt {5}}\, x}{2}, \frac {\sqrt {-6+3 i \sqrt {5}}}{3}\right )\right )}{5 \sqrt {2-2 i \sqrt {5}}\, \sqrt {-3 x^{4}+2 x^{2}-2}\, \left (2+2 i \sqrt {5}\right )}\)	\(238\)

Input:

int(1/(-3*x^4+2*x^2-2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/20*x*(3*x^2+4)/(-3*x^4+2*x^2-2)^(1/2)-3/5/(2-2*I*5^(1/2))^(1/2)*(1-(1/2 
-1/2*I*5^(1/2))*x^2)^(1/2)*(1-(1/2+1/2*I*5^(1/2))*x^2)^(1/2)/(-3*x^4+2*x^2 
-2)^(1/2)*EllipticF(1/2*(2-2*I*5^(1/2))^(1/2)*x,1/3*(-6+3*I*5^(1/2))^(1/2) 
)+6/5/(2-2*I*5^(1/2))^(1/2)*(1-(1/2-1/2*I*5^(1/2))*x^2)^(1/2)*(1-(1/2+1/2* 
I*5^(1/2))*x^2)^(1/2)/(-3*x^4+2*x^2-2)^(1/2)/(2+2*I*5^(1/2))*(EllipticF(1/ 
2*(2-2*I*5^(1/2))^(1/2)*x,1/3*(-6+3*I*5^(1/2))^(1/2))-EllipticE(1/2*(2-2*I 
*5^(1/2))^(1/2)*x,1/3*(-6+3*I*5^(1/2))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.63 \[ \int \frac {1}{\left (-2+2 x^2-3 x^4\right )^{3/2}} \, dx=\frac {\sqrt {-2} {\left (3 \, x^{4} - 2 \, x^{2} + \sqrt {-5} {\left (3 \, x^{4} - 2 \, x^{2} + 2\right )} + 2\right )} \sqrt {\frac {1}{2} \, \sqrt {-5} + \frac {1}{2}} E(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {-5} + \frac {1}{2}}\right )\,|\,-\frac {1}{3} \, \sqrt {-5} - \frac {2}{3}) + \sqrt {-2} {\left (3 \, x^{4} - 2 \, x^{2} - 3 \, \sqrt {-5} {\left (3 \, x^{4} - 2 \, x^{2} + 2\right )} + 2\right )} \sqrt {\frac {1}{2} \, \sqrt {-5} + \frac {1}{2}} F(\arcsin \left (x \sqrt {\frac {1}{2} \, \sqrt {-5} + \frac {1}{2}}\right )\,|\,-\frac {1}{3} \, \sqrt {-5} - \frac {2}{3}) + 2 \, \sqrt {-3 \, x^{4} + 2 \, x^{2} - 2} {\left (3 \, x^{3} + 4 \, x\right )}}{40 \, {\left (3 \, x^{4} - 2 \, x^{2} + 2\right )}} \] Input:

integrate(1/(-3*x^4+2*x^2-2)^(3/2),x, algorithm="fricas")
 

Output:

1/40*(sqrt(-2)*(3*x^4 - 2*x^2 + sqrt(-5)*(3*x^4 - 2*x^2 + 2) + 2)*sqrt(1/2 
*sqrt(-5) + 1/2)*elliptic_e(arcsin(x*sqrt(1/2*sqrt(-5) + 1/2)), -1/3*sqrt( 
-5) - 2/3) + sqrt(-2)*(3*x^4 - 2*x^2 - 3*sqrt(-5)*(3*x^4 - 2*x^2 + 2) + 2) 
*sqrt(1/2*sqrt(-5) + 1/2)*elliptic_f(arcsin(x*sqrt(1/2*sqrt(-5) + 1/2)), - 
1/3*sqrt(-5) - 2/3) + 2*sqrt(-3*x^4 + 2*x^2 - 2)*(3*x^3 + 4*x))/(3*x^4 - 2 
*x^2 + 2)
 

Sympy [F]

\[ \int \frac {1}{\left (-2+2 x^2-3 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- 3 x^{4} + 2 x^{2} - 2\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(-3*x**4+2*x**2-2)**(3/2),x)
 

Output:

Integral((-3*x**4 + 2*x**2 - 2)**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (-2+2 x^2-3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-3 \, x^{4} + 2 \, x^{2} - 2\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(-3*x^4+2*x^2-2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((-3*x^4 + 2*x^2 - 2)^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (-2+2 x^2-3 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-3 \, x^{4} + 2 \, x^{2} - 2\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(-3*x^4+2*x^2-2)^(3/2),x, algorithm="giac")
 

Output:

integrate((-3*x^4 + 2*x^2 - 2)^(-3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (-2+2 x^2-3 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-3\,x^4+2\,x^2-2\right )}^{3/2}} \,d x \] Input:

int(1/(2*x^2 - 3*x^4 - 2)^(3/2),x)
 

Output:

int(1/(2*x^2 - 3*x^4 - 2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (-2+2 x^2-3 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-3 x^{4}+2 x^{2}-2}}{9 x^{8}-12 x^{6}+16 x^{4}-8 x^{2}+4}d x \] Input:

int(1/(-3*x^4+2*x^2-2)^(3/2),x)
 

Output:

int(sqrt( - 3*x**4 + 2*x**2 - 2)/(9*x**8 - 12*x**6 + 16*x**4 - 8*x**2 + 4) 
,x)